Closed-form inverse kinematics equations may be derived for many types of manipulators which make the inverse kinematics solutions for the joint variables for these manipulators relatively simple to calculate.
With 6 degree of freedom (DOF) manipulators, closed-form inverse kinematics equations can often be derived when any three adjacent joints of the manipulator are either intersecting or parallel to one another. The theory for 7 DOF manipulators is not as well understood, however it is possible to derive closed-form equations for many manipulators which resemble the human arm.
Non-closed-form methods for determining joint coordinates of a manipulator via the inverse kinematics are also known. Generally all of these methods involve numerically iterating to within a maximum allowable error of the desired solution, and are usually intended to be used on manipulators which may not have closed-form inverse kinematics equations.
Wang and Chen describe a combined optimization method which they state can be used on any manipulator geometry (see Int. J. Robot. Res., 7(4):pages 489-499, August 1991). This method uses the cyclic coordinate descent method to find an initial estimate of the desired solution and then uses the Broyden-Fletcher-Shanno variable metric method to find a solution which is within the maximum tolerable error. The method is applicable to manipulators of arbitrary number of degrees of freedom, has guaranteed global convergence, and is not sensitive to manipulator singularities.
Tsai and Morgan in Trans. ASME J. Mech. Transmiss., Automat. Des., 107(2): 189-200, June 1985 describe a method which is applicable to general six- and five-DOF manipulators and which can generate all possible solutions for a desired endpoint position and orientation. This was achieved by converting the problem into a system of 8 second order equations in 8 unknowns and then solving this system for all possible solutions, by continuation methods.
Lee and Liang in Mech. Mach. Theory, 23(7): 219-226, 1988 describe a general method for solving the inverse kinematics problem for 6 DOF manipulators with rotational joints. This method involves deriving a 16th order polynomial equation in the tan-half-angle of the first joint (.theta..sub.1), and then (for each desired endpoint position and orientation) numerically solving for all of the zeroes of this polynomial equation. All the real roots obtained represent a viable joint angle value for .theta..sub.1 while the other five joint variable values are calculated using closed-form equations which are functions of .theta..sub.1.
Takano describes an inverse kinematics method for 6 DOF manipulators which requires closed-form Inverse kinematics equations for the first 3 joints in terms of the endpoint position of the third line (see Journal of the Faculty of Engineering, The University of Tokyo (B), 38(2): 107-135, 1985. It also requires closed-form inverse kinematics equations for the last 3 links in terms of the orientation of the final link. Endpoint position and endpoint orientation are then solved for separately, one after the other in an iterative manner, until convergence is obtained to within the required accuracy.
Manseur and Doty in Int. J. Robot. Res., 7(3): 52-63, June 1988 teach a fast algorithm which can be used with 6 DOF manipulators that have revolute joints at the first and last links, and for which one is able to derive closed-form equations for joint variables 2 through 6 as a function of joint variable 1. Under these conditions, it is shown that a single non-linear equation can be generated with the only unknown being joint variable 1. This equation is then solved using the Newton-Raphson method.
In IEEE Trans. Robotics and Automat., 5(5): 555-568, October 1989 Tourassis and Ang describe a fast method which is applicable to general 6 DOF geometries. The method is based on creating a system of 3 non-linear equations in 3 unknowns which represent the Cartesian position of the wrist. Conditions under which the method will converge are given and the desired manipulator configuration can be easily specified. The algorithm is also applicable for real time use.
Other methods which have been extensively used to solve the inverse kinematics problem are called Inverse Jacobian Methods and are based upon Newton algorithms which are used to solve systems of non-linear equations. The non-linear equations are usually derived from expressions in which the desired endpoint position and orientation are functions of the manipulator's joint variables (i.e. forward kinematics equations). These methods work by linearizing the system of equations using a Taylor's series expansion around the manipulator's initial position (i.e. calculating the Jacobian of the forward kinematics equations). This system of linear equations, when solved, provides an approximation to the joint variable values which give the desired endpoint position and orientation. Inverse Jacobian methods are generally applicable to all manipulator geometries, including redundant manipulators, and can be suitable for real time control.
Some Newton-based algorithms use damped least-squares methods to compensate for problems that are associated with Jacobian singularities. Nakamura in his PhD thesis "Kinematical Studies on the Trajectory Control of Robot Manipulators", Kyoto, Japan, June 1985 describes a method wherein damping factor is increased as the manipulator approached a Jacobian singularity region. U.S. Pat. No. 4,893,254 issued Jan. 9, 1990 to Chan et al describes a system wherein the damping factor is a function of residual error to provide inverse kinematics control of manipulators near singularity regions. Hutchinson in his master thesis "Manipulator inverse kinematics based recursive least squares estimation", Dept. of Electrical Eng., University of British Columbia, December 1988 showed that a least-squares approximated Jacobian used in a Newton-Raphson type algorithm has similar convergence properties to the damped least-squares method.
Novakovic and Nemec in "A solution to the inverse Kinematics Problem Using the Sliding Mode", IEEE Trans.Robotics and Automat, 6(2): 247-252, April 1990 developed an algorithm based on sliding mode control and Lyapunov theory. The algorithm is tested on a 4 axis manipulator which has no known closed-form solutions to its inverse kinematics. The method is indicated as being computationally more efficient than Newton based methods and not as prone to singularity problems.
Powell "A Hybrid Method for Nonlinear Equations". In P. Rabinowitz, editor, "Numerical Methods for Nonlinear Algebraic Equations, pages 87-114. London, UK: Gordon and Breach, 1970 developed a hybrid algorithm for solving systems of linear equations which uses the Newton-Raphson method, the gradient descent method, and a combination of the two. The combination of these three algorithms give Powell's hybrid algorithm, in general, better convergence properties than most other non-linear equation solving algorithms. This inverse kinematics formulation is applicable to general manipulator geometries, allows real time control, and has better convergence properties than methods that are based solely on the Newton-Raphson approach.
Another approach to the inverse kinematics problem involves the integration of joint velocities which are obtained using inverse Jacobian techniques. Tsai and Orin "A Strictly Convergent Real-time solution for Inverse kinematics of Robot Manipulators", J. Robotic Syst., 4(4): 477-501, August 1987 used a variation of this approach along with a special-purpose inverse kinematics processor to obtain a system that works in real time. This method has good convergence properties in that it usually only fails to converge in manipulator singularity regions.
None of the above methods simultaneously
1. guarantees convergence to a desired endpoint position and orientation if it exists; PA0 2. is applicable to real-time manipulator control; PA0 3. gives direct control over the manipulator's configuration; and PA0 4. is applicable to manipulators of arbitrarily many degrees of freedom.